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A generalization of the Auslander-Nagata
purity theorem

Author: Miriam Ruth Kantorovitz
Journal: Proc. Amer. Math. Soc. 127 (1999), 71-78
MSC (1991): Primary 13B15; Secondary 13B02
MathSciNet review: 1458881
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $B \hookrightarrow A$ be a module finite extension of normal domains. We show that if $B \hookrightarrow A$ is unramified in codimension one and if $A$ has finite projective dimension over $B$, then $A$ is étale over $B$. Our proof makes use of P. Roberts' New Intersection Theorem.

References [Enhancements On Off] (What's this?)

  • [Ab] Shreeram Abhyankar, Ramification theoretic methods in algebraic geometry, Annals of Mathematics Studies, no. 43, Princeton University Press, Princeton, N.J., 1959. MR 0105416
  • [Au] Maurice Auslander, On the purity of the branch locus, Amer. J. Math. 84 (1962), 116–125. MR 0137733
  • [AB] M. Auslander and D. A. Buchsbaum, On ramification theory in noetherian rings, Amer. J. Math. 81 (1959), 749–765. MR 0106929
  • [Bor] Adam Borek, Weak purity for Gorenstein rings, J. Algebra 175 (1995), no. 2, 409–450. MR 1339650, 10.1006/jabr.1995.1195
  • [Bou] Nicolas Bourbaki, Commutative algebra. Chapters 1–7, Elements of Mathematics (Berlin), Springer-Verlag, Berlin, 1989. Translated from the French; Reprint of the 1972 edition. MR 979760
  • [BH] Winfried Bruns and Jürgen Herzog, Cohen-Macaulay rings, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR 1251956
  • [Ch] Wei Liang Chow, On the theorem of Bertini for local domains, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 580–584. MR 0096667
  • [Cu] Steven Dale Cutkosky, Purity of the branch locus and Lefschetz theorems, Compositio Math. 96 (1995), no. 2, 173–195. MR 1326711
  • [F] Robert M. Fossum, The divisor class group of a Krull domain, Springer-Verlag, New York-Heidelberg, 1973. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 74. MR 0382254
  • [G1] Phillip Griffith, Normal extensions of regular local rings, J. Algebra 106 (1987), no. 2, 465–475. MR 880969, 10.1016/0021-8693(87)90008-1
  • [G2] Phillip Griffith, Some results in local rings on ramification in low codimension, J. Algebra 137 (1991), no. 2, 473–490. MR 1094253, 10.1016/0021-8693(91)90102-E
  • [SGA2] Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (𝑆𝐺𝐴 2), North-Holland Publishing Co., Amsterdam; Masson & Cie, Éditeur, Paris, 1968 (French). Augmenté d’un exposé par Michèle Raynaud; Séminaire de Géométrie Algébrique du Bois-Marie, 1962; Advanced Studies in Pure Mathematics, Vol. 2. MR 0476737
    Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux. Fasc. I: Exposés 1–8; Fasc. II: Exposés 9–13, Séminaire de Géométrie Algébrique 1962. Troisième édition, corrigée. Rédigé par un groupe d’auditeurs, Institut des Hautes Études Scientifiques, Paris, 1965 (French). MR 0210718
  • [HW1] Craig Huneke and Roger Wiegand, Tensor products of modules and the rigidity of 𝑇𝑜𝑟, Math. Ann. 299 (1994), no. 3, 449–476. MR 1282227, 10.1007/BF01459794
  • [HW2] C. Huneke and R. Wiegand, Tensor products of modules, rigidity, and local cohomology, submitted.
  • [Ma] Hideyuki Matsumura, Commutative ring theory, 2nd ed., Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1989. Translated from the Japanese by M. Reid. MR 1011461
  • [Mi] C. Miller, Hypersurface sections: A study of divisor class groups and the complexity of tensor products, Ph. D. thesis, University of Illinois at Urbana-Champaign, 1996.
  • [N1] Masayoshi Nagata, On the purity of branch loci in regular local rings, Illinois J. Math. 3 (1959), 328–333. MR 0106930
  • [N2] Masayoshi Nagata, Local rings, Interscience Tracts in Pure and Applied Mathematics, No. 13, Interscience Publishers a division of John Wiley & Sons New York-London, 1962. MR 0155856
  • [R] Paul Roberts, Le théorème d’intersection, C. R. Acad. Sci. Paris Sér. I Math. 304 (1987), no. 7, 177–180 (French, with English summary). MR 880574
  • [W] Keiichi Watanabe, Certain invariant subrings are Gorenstein. I, II, Osaka J. Math. 11 (1974), 1–8; ibid. 11 (1974), 379–388. MR 0354646
  • [Z] Oscar Zariski, On the purity of the branch locus of algebraic functions, Proc. Nat. Acad. Sci. U.S.A. 44 (1958), 791–796. MR 0095846

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Additional Information

Miriam Ruth Kantorovitz
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

Keywords: Auslander-Nagata purity, unramified extension
Received by editor(s): October 17, 1996
Received by editor(s) in revised form: May 14, 1997
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1999 American Mathematical Society